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American Journal of Physics

Transmission and scattering by an absorptive potential

A. Uma Maheswari, Sidhaarth Mahadevan, P. Prema, C. S. Shastry, S. K. Agarwalla

Published Feb 12, 2007• 2 citations
Publisher

Abstract

Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.

DOI
10.1119/1.2388972

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Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.
Transmission and reflection across a rectangular barrier at energies below and above barrier is the most commonly studied topic in non-relativistic quantum mechanics. However the subtle inter-relationship between the barrier problem and the corresponding well problem is not widely known, in particular when absorption is present. In this article we show that when a particle traverses an absorptive medium, at any given energy the absorption peaks for a particular value of absorption potential strength W0. Similarly we study the corresponding cases when incident energy E is increased keeping W0 constant. Further, we show that for a given E when W0. is made very large absorption gradually decreases and reflection overtakes it and tends towards unity. We also study the case of transmission across a potential barrier and well and interpret physically the behavior of absorption, transmission and reflection.
Strongly singular potentials in one dimension are analyzed, namely the n th derivative interaction     x n  .For the repulsive case, the reflection and transmission coefficients are derived.It is shown that for n = even, these coefficients satisfy the unitarity of the scattering matrix for all values of the incident energy and the barrier becomes perfectly reflective for a specific value of the energy.For n=odd 1  , however, the coefficients do not satisfy the unitarity of the scattering matrix except for one specific value of the incident energy at which the barrier becomes perfectly reflective.For the n=1 case, the results showed that the coefficients satisfy the unitarity of the scattering matrix.For the attractive case, both the bound states and the scattering states are examined.It is shown that, for n=even 0  , there exist two bound states and only one bound state for n=0.For the scattering states, it is demonstrated that one recovers the same coefficients that were obtained for the repulsive case.
Applying the commonly used partial wave method in the quantum theory of collisions, the scattering cross-sections of the two- and three-dimensional attractive and repulsive potentials are expressed as sums of phase shifts. With the help of the Maple software program, the phase shifts and cross-sections are plotted as functions of the kinetic energy and potential height, and it is demonstrated that the systematic convergence of the cross-sections as higher angular-momentum phase shifts are included in the summations. It is found that the cross-sections of the attractive potential oscillate with both the kinetic energy and the potential strength, even though the phase shifts are monotonic. It is also shown that in the case of the repulsive potential, both the phase shifts and cross-sections change profoundly when the kinetic energy of the incident particle is lower than the potential height.
The scattering properties of one-dimensional potential with gain are studied by using a Schrödinger-like equation. The corresponding Hamiltonian is non-Hermitian with a real energy spectrum. The amplification-absorption duality previously observed is interpreted in terms of the transmission and reflection phases. For a rectangular barrier, the transmission phase oscillates with the barrier width as for passive systems, but the oscillations period is significantly reduced in the absorption region. In this region the reflection phase vanishes and the multiple scattering and interferences dominate. The gain effect is also investigated for double barrier structures as well as superlattices with active potentials. It is found that resonant tunneling energy and the mini-band width are not influenced by the gain, but the transmission is enhanced for small values of the potential imaginary part. For large values, the resonant transmission significantly decreases and the mini-bands disappear.
Abstract A familiar classroom heuristic for hard-sphere scattering is that the low-energy cross section ‘equals the surface area’ because a long-wavelength particle can ‘feel’ the sphere from all directions. More precisely, this heuristic means that, when the wavelength is large compared with the size of the scatterer, the wave is sensitive to the global boundary condition imposed by the whole hard obstacle; in the zero-energy limit this global response is represented by the monopole term of an exterior Laplace problem. While useful as intuition, the heuristic prompts a natural question for students: is there a general connection between surface area and the threshold cross section of a non-spherical hard target? This pedagogical note, aimed at advanced undergraduates and beginning graduate students, shows how to answer that question systematically. For a perfectly hard obstacle the zero-energy Schrödinger equation reduces to an exterior Laplace problem with Dirichlet boundary conditions. Comparing its far-field monopole term with the corresponding electrostatic problem yields a direct mapping between the scattering length and the electrostatic capacitance of the same shape. The mapping provides a simple route to threshold cross sections for bodies of arbitrary shape, without relying on separability. Using published capacitances for several examples, we compare the threshold cross section with surface area and show that the equality for a sphere is not a general surface-area law.